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The conformal map $z\to {z}^{2}$ of the hodograph plane. (English) Zbl 0806.65005

The author studies the map of a curve $z\left(t\right)=u\left(t\right)+iv\left(t\right)$ onto $w\left(t\right)=\int {z}^{2}\left(s\right)ds$ with appropriately chosen starting point for the integration. The latter curves are shown to have very convenient properties for differential-geometric computation since ${w}^{\text{'}}=\left({u}^{2}-{v}^{2}\right)+2uvi$. If $u$, $v$ are polynomials of degree $n$, the components of $w$ are polynomials of degree $2n+1$. Special attention is given to Bézier control of the curves $w$.

The author complains about the lack of books using complex methods for real geometry. To his list of books should be added the classic “Inversive geometry” by F. Morley and F. V. Morley (1933; Zbl 0009.02908) and more recent books by J. L. Kavanau [e.g. Structural equation geometry (1983; Zbl 0541.51001) and Curves and symmetry (1982; Zbl 0491.51001)].

##### MSC:
 65D17 Computer aided design (modeling of curves and surfaces) 51N20 Euclidean analytic geometry